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IAA Transactions, No. 8, ``Celestial Mechanics'', 2002
Superosculating intermediate orbits and their
applications to study the perturbed motion
V. A. Shefer
Research Institute of Applied Mathematics and Mechanics, Tomsk State
University, Tomsk, Russia
A method of construction of intermediate orbits for approximating the real mo­
tion of celestial bodies in the initial part of trajectory is described. The method
is based on introducing a fictitious attracting centre and is used to generalize
Batrakov's approach [1]. The main idea of the method, proposed and realized in
earlier publications [2,3] for the first time, lies in the fact that the gravitational pa­
rameter of the fictitious centre for the initial part of motion is not constant but is
determined as a function of time from the condition minimizing the perturbations.
The motion along the intermediate trajectory about the fictitious centre is not
Keplerian. In the present work it is described by the equations of the perturbed
version of the Gylden--Meshchersky problem. This generalizes our approach as
compared with the previous papers [2, 3], in which we have used the equations of
the Gylden--Meshchersky problem in their classical form [4]. Here, we consider a
case where the gravitational parameter defining the intermediate motion varies
in accordance with the Eddington--Jeans mass--variation law. New classes of or­
bits having second-- and third--order tangency to the perturbed trajectory at the
initial instant of time are constructed. For planar motion, the tangency increases
by one order. The constructed intermediate orbits approximate the perturbed
motion better than the osculating Keplerian orbit and analogous orbits of other
authors.
The applications of the orbits constructed in Encke's method [5] for special
perturbations and in the procedure for predicting the motion in which the per­
turbed trajectory is represented by a sequence of short arcs of the intermediate
orbits are suggested. The first application leads to generalized Encke's algorithms.
It results also in new methods for solving the equations of orbital motion whose
accuracy order coincides with the tangency order of the used intermediate orbit.
The application of the Runge rule and Richardson extrapolation [6] to the latter
allow us to obtain methods of higher orders. The new methods have been com­
pared with the classical Encke method and numerical integration of the equations
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of motion by the well--known Runge--Kutta--Fehlberg methods of the 4th and 7th
orders [6]. The comparisons have been performed in computing the perturbed
orbits of some small bodies of the Solar system.
The constructed intermediate orbits can be effectively applied also in the
problem of determining preliminary orbits of celestial bodies from observations
and their subsequent improvement.
References
1. Batrakov Yu. V. Intermediate Orbits Approximating the Initial Part of Pe­
rurbed Motion. Bull. ITA, 1981, 15, 1--5 (in Russian).
2. Shefer V. A. Superosculating Intermediate Orbits for the Approximation of
Perturbed Motion. Second--Order Tangency. Astron. Rep., 1998, 42, 837--
844.
3. Shefer V. A. Superosculating Intermediate Orbits for the Approximation of
Perturbed Motion. Third--Order Tangency. Astron. Rep., 1998, 42, 845--854.
4. Meshchersky I. W. Ein Specialfall des Gyldenschen Problemes. Astron.
Nachr., 1893, 132, 129--130.
5. Encke J. F. Ё
Uber die Berechnung der speciellen StЁorungen. Berliner As­
tronomisches Jahrbuch fЁur 1857, 1855, 307--335.
6. Hairer E., Norsett S. P., Wanner G. Solving Ordinary Differential Equations.
Nonstiff Problems. Berlin, Heidelberg: Springer Verlag, 1987.
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